Sound Absorption in Solids

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Sound Absorption in Solids Slonimskii, G. L.

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T i t l e :

Author:

R e f e r e n c e :

ITATITO?TAL RESEARCH COUITCYL O F CANADA

Tir _{: hiucal ? I r a l ~ r l a t ion TT-,641}

Zhw, Eicsp, i T e o r e t , Fiz, 7 (12): 1457-1462,

19370

SOUND ABSORPTION IN SOLIDS Summary

The

absorption of plane sound waves in a solid body resulting from the incid:ent wave splitting into two waves of different freauencies is considered, This Drocess is due to the presence

6f

cubic terms in expressions ?or the den- sity of elastic energy regarded as perturbation, The effect appears to be possible (in the first approximation) only for longitudinal incident waves,

In the work of L. Landau and G, Rumer the absorption of sound in a solid body is regarded as the result of collisions be- tween sound quanta and heat quanta, Here it is found that by vir- tue of conservation laws, in the first approximation it is only possible to have absorption of transverse sound waves by longitudi- nal heat waves. The absorption of longitudinal waves is possible, according to this theory, only in the second approximation, taking into account quartic terms in expressions for the density of elas- tic energy and when at least four waves are involved,

Meanwhile, as we will show, if one considers that the sound quantum splits into two quanta, which is possible under non- linear conditions, then, contrary to the opinion expressed above, by virtue of the conservation laws it is possible for the longi-

tudinal sound quantum to split and impossible for the transverse sound quantum.

In

fact the laws of the conservation of energy and momen-

tum must

hold, i.e,,

Moreover, the theory of elasticity requires that the propagation speed of longitudinal waves cC should be greater than

-

'Jh~hus xe have ko 4 k,

+

Ic2, where ko =

1 ~ ~ 1 ,

Ic, =ik !-..I i

'

c 2 =

,

and n l s o ko = w0/co, k , = w,/c,, k, = u,/c,, hence

Consider t h e p o s s i b l e cases.

1. c O = c , , c l = C , ~ C , = C ~ . Then

wo/cc

s

o,/c, + 02/ct. This i r l c r ~ u a l i t y h o l d s only when ct > ct.

2. co = cCY c I = c t , c 2 = ct. Then uo/c4 4 q c t + u2/ct.

T h i s a l s o i s p o s s i b l e only when ce > ct.

Consequently t h e a b s o r p t i o n of lorei-l;!ldin:?l m:;ver; occurs i n t h e :?ir:;,i; ci;~proxiinz~tion because of' s p l i t t i n g and only 1 1 1 'i'ele

secozi4 51:;proxj.~?clat i o n because of z 3 s o r p t i o n by h e a t ~u::lnLa.

'Pie ~vill n o t e t h a t t h e tr<ititsverze waves, .:.s i . t i s easy t o s e e from t n e cojnseravation laws, do n o t s p l i t i n t31c Pirsi; ;:p;)120j;i-

nation.

In

f;s;.\=t , y j . . l l c o n s i d e r t h e p o s s i b l e c:?ss.s (1:;; $;.;c s p l i i ; t i n s of' l a t e r a l quanta: 2. Co = Ct, C , = Ce, e = ceo 2 Then oo/ct 4 ,/cc + o*/ce.

In 5o-Lli c a s e s t h e i n e i ~ ~ a l i t i e s obtained hold. 0111.3: x'ilcn

c, > cC :.';2ich con'ir:lclic-ts t h e

elastic it:^

theory. b

Note that in this work we are conrpletely neglecting sound dispersion and therefore we are not considering the splitting of the transverse sound quantum into two transverse quanta and the longitudinal quantum into two longitudinal quanta. These effects take place only when dispersion is taken into account, since then the quanta obtained after splitting will have a velocity somewhat different from the velocity of the initial q u a n t m

Strictly speaking one should take into account not only the splitting of the sound quantumbut also the reverse process of reformation, i, e., the fomnation of one quantum from two. We

neglect the second process since, at low temperatures, heat vi- brations are practically not excited in a solid body.

Like the work by Landau and Rumer the present work refers to short waves and therefore it cannot be experimentally proven at the present time*

The calculation proceeded as follows:

A

plane sound wave with a wave vector

J50

and a frequency

o propagates in

an

isotropic medium, When the classical theory

0

of elasticity is used it is impossible to have splitting*

If,

in the expressions for energy density, cubic terms are considered we have the terminal probability of splittinga These cubic terms in expressions for energy density we regard as perturbation causing splitting of the sound wave (the wave vector

so

andfrequency coo) into two sound waves (wave vectors

&,

and lc2 and frequencies m, and

W * '

We restrict ourselves to the first approximations This means that only three waves will be considered,

As

shown above, in

the presence of three waves it is possible to have splitting only of the longitudinal wave,

Terms in the second set OF brackets we regard as pertur- bation resulting in splitting of the propagating wave,

Transition Probability

We represent sound waves in the form of the sum of har-

monic waves : .-

where is the unit vector of the direction of polarization, a, is the amplitude, &a is the wave vector, Let the number of sound

quanta be N,. We want to find the probability of transition from the state with the nunibers of sound quanta No, N,, N2 into states with numbers of sound quanta of No

-

1, N,

+

1, N,

+

1.

Substituting in the expression for the density of pertur- bation energy

where the indices l and 2 refer to waves obtained after splitting, we find

(1) v(2) y(o) + .u (2)v(o) ( 1 ) ) (0) v(l) v(2) +

+ Uaa a0 a0 aa ap Vafi

-

2B(Uap py ya

+

u

( 1 ) (2) v(o) + u (2) v(o)

v(

₎

v~~

Ya a@ PY 'ya

In

this expression all terms are discarded which contain functioiis with one sign two or three times since they are of no significance in our calculation,

From the theory of harmonic oscillators we know the

matrix elements of the displacement, Only the following elements are non-zero:

where _rn is the mass of the volume

V.

The matrix elements of the components

u

and

v

we find

a@ a@

by differentiating the displacement component:

where ea, ka denote

x,

y, z

-

the components of the vectors &and

For longitudinal waves

[a&]

= O9 consequently

For transverse waves ( g g ) = 0 consequently

Therefore in case No, 1 = 0,

[ftlk,]

= 0,

(

)

=

0) the expression for the density of perturbation energy can be presented in the form:

In

case No. 2 ([dodo e k

1

= 0,

(plkl)

= 0,

(p2&k)

= 0) the density of perturbation energy has the form

In further calculations we will compute the following ex- pression:

1

q, =

-

(elakla

+

elak,a) (e,ak,@

+

e,pk2a) (eoakoB + eo@koa), 8

For case No. 1 we get: _{ql = kl}

_(s2eo)

_{($so), q, =} ko(&,r2)

(gl$),

1

r = -

kokl

I(eoel)

(el%)

(

)

+

(

)

(soel)

(~,$~)l.

2

For case No. 2 we get: 1

s,

=

-

k0

I(ele2)

+

( 5 , ~ ~ )

(klg2)lt

Calculating all u and

v

and substituting them in the

aP

expression for the density of perturbation energy we find:

Hence we find that the matrix element we are interested in has the form

where

g = s o

-

3 ,

- S z 9

O = w - w

-

o

.

0 3 2

Integrating over the volume V we find the matrix of the perturbation energy Hperturl,

0

Note that e-i(&)d%

I

is non-zero only under the condition K = 0, i,e,, only when the law of the conservation of monlentum holds,

As a result of integration we find

For calculating the probability of excitation of a given state in a unit of time we have the known formula of the perturb-, ation theory

where a, is the probability amplitude.

For our case (an ( 0 ) = 0,

n

n n f NoN1N2; _~N,N,N,

0 1% 0.1 2

( 0 ) = 1) this formula takes on the following simpler form:

Wo gives the probability of transition in a unit of time of the longitudinal sound quantum of the frequency coo into two sound quanta of the frequencies w, and w, of a given polarization.

To get the transition probability of a sound quantum into two quanta of given frequencies but with arbitrary polarization it is necessary to integrate the expression for Wo over angles

Fig* 1 Case NO, 1.

Fig, 2 Case No, 2.

We have (see Figs. 1 and 2)

Carrying out the integration, assuming that

N,

= N, = 0

and dividing both sides of the equation by No we find, for the

probability of the decomposition of a sound quantum into two quanta, the expression,

where

~ ~ ( e ~ ,

e2)={Q[sin 2e2

+

sin 2(e1+

e2)]

+ 3R cos

el

+ 2e2)j2,

The nurriber of states at which

&,

is directed towards the element of the solid angle ~$2, and the absolute magnitude of k, lies in the range between k, and k,

+

dk,, is given by the known formula

The full required probability of splitting in a unit of time is given by the integral,

Carrying out this integration (taking into account the fact that k,

,

k,,

el,

8 , are connected among each other by con- servation laws) gives

where c = c4/ct and

u

= (c

-

l)/(c

+

1

)

.

For c = e w e have:

Thus

longitudinal sound quanta

are

absorbed 'because of splitting into longitudinal and transverse quanta and because of splitting into two lateral sound quanta. Thus the probability of the absorption of a longitudinal sound quantum owing to any

splitting can be found by combining the probabilities of splitting found for cases 1 and 2.

For the value of c =-we find that only splitting into two transverse quanta is significant.

In

conclusion

I

would like to thank Professor Iu. Rumer under whose direction this work was carried out,

Moscow

Faculty of Physics, Moscow State University.

Reference

1. Landau,, L. and Rumer, G. h e r Schallabsorption

in

festen Korpern (sound absorption in solids). Sow. Phys. 11